The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric function. The period of change is exactly $1$ year. The temperature peaks around July $26$ at $78^\circ F$, and has its minimum half a year later at $49^\circ F$. Assuming a year is exactly $365$ days, July $26$ is $\dfrac{206}{365}$ of a year after January $1$. Find the formula of the trigonometric function that models the daily low temperature $T$ in Guangzhou $t$ years after January $1$, $2015$. Define the function using radians. $ T(t) = $
Solution: Let's start by writing a formula for the temperature $u$ years after July $26$. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. At $u = 0$, on July $26, 2014$, the temperature will be at its peak of $78^\circ F$. So we'll use a cosine function, since cosine functions reach their peak at $0$. The temperature in Guangzhou has period $1$ year. Its midline is halfway between its maximum and minimum temperatures, or $\dfrac{78 + 49}{2} = 63.5$ Its amplitude is half the difference between its maximum and minimum temperatures, or $\dfrac{78 - 49}{2} = 14.5$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we can stretch it horizontally by a factor of ${\dfrac{1}{2\pi}}$, stretch it vertically by a factor of ${14.5}$, and move it up ${63.5}$ units: $ T(u) = {14.5}\cos\left({2\pi}u\right) + {63.5}$ Since July $26$ is $\dfrac{206}{365}$ of a year after January $1$, The day that is $t$ years after January $1$ is $t - \dfrac{206}{365}$ years after July $26$. $ T(t) = {14.5}\cos\left({2\pi}\left(t - \dfrac{206}{365}\right)\right) + {63.5}$ The function $ T(t) = {14.5}\cos\left({2\pi}\left(t - \dfrac{206}{365}\right)\right) + {63.5}$ has period $1$, amplitude $14.5$, and midline $y = 63.5$, and reaches its peak at $\dfrac{206}{365}$, so it's a good model of the temperature in Guangzhou.